Publications
PUBLISHED / ACCEPTED
Tranched graphs: consequences for topology and dynamics
Authors: Michał Kowalewski, Piotr Oprocha
Fundamenta Mathematicae 273, 1-46 • 2026
We compare quasi-graphs and generalized sin(1/x)-type continua, which are two classes of continua that generalize topological graphs and contain the Warsaw circle as a nontrivial common element. We show that neither class is a subset of the other, provide some characterizations, and present illustrative examples. We unify both approaches by considering the class of tranched graphs, compare it to concepts known from the literature, and describe how the topological structure of its elements restricts possible dynamics.
Every nondegenerate Peano continuum admits a pure mixing selfmap
Authors: Klara Karasova, Michał Kowalewski, Piotr Oprocha
to appear in Proceedings of the American Mathematical Society • 2026
We prove that every Peano continuum (a space that is a continuous image of [0,1]) admits a topologically mixing but not exact map. The constructed map has a dense set of periodic points.
PREPRINTS
Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors
Authors: Magdalena Foryś-Krawiec, Jana Hantáková, Michał Kowalewski, Piotr Oprocha
preprint arXiv:2511.09718 •
We study the observable long-term behavior of typical continuous dynamical systems on the interval [0,1]. For a residual subset of C([0,1]), the Milnor, statistical, and physical (in the sense of Ilyashenko) attractors coincide and are equal to the non-wandering set. This unified attractor governs the time-averaged dynamics of almost all initial conditions and depends continuously on the map with respect to the Hausdorff metric. From the physical viewpoint, it represents the ensemble of observable steady states describing the long-term statistical behavior of the system. Nevertheless, it is not Lyapunov stable and contains no dense orbits, implying the generic absence of Palis attractors. Thus, generic continuous dynamics admit a well-defined observable attractor even when all classical mechanisms of stability fail, showing how observable statistical behavior persists in the absence of SRB measures or hyperbolic structure.